The Konus-Wind Catalog of Gamma-Ray Bursts with Known Redshifts. I. Bursts Detected in the Triggered Mode

4.1.1. Analysis

4.1.2. Results

Table 2 summarizes the results of our temporal and lag analyses. The first column contains the GRB name (see Table 1). Next, the values of T₁₀₀, T₉₀, and T₅₀ are listed along with the corresponding start times t₀, t₅, and t₂₅ given relative to the trigger time T₀. For GRB 081203A, which was detected during the data output of GRB 081203B, no high-resolution light curves are available and, thus, only a rough estimate of T₁₀₀ is provided. While for weak KW GRBs, T₁₀₀ and T₉₀ are nearly similar measures of duration (Figure 3), for brighter bursts, T₁₀₀ becomes more sensitive to the existence of weak precursors or extended tails. This behavior is particularly apparent for such remarkable events as the "naked-eye" GRB 080319B (Racusin et al. 2008); the ultra-luminous GRB 110918A (Frederiks et al. 2013); the nearby, ultra-bright GRB 130427A (Maselli et al. 2014); and two recent highly energetic events, GRB 160623A (Frederiks et al. 2016) and GRB 160625B (Svinkin et al. 2016a; Zhang et al. 2016). The latter burst features a precursor separated from the main episode by a long interval of quiescence and the four former bursts are characterized by slowly decaying tails of hard X-ray emission that were bright enough to be detected in the KW G2 band for hundreds of seconds.

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Table 2.  Durations and Spectral Lags

Burst	t0	T100	t5	T90	t25	T50	${\tau }_{\mathrm{lagG}2{\rm{G}}1}$	${\tau }_{\mathrm{lagG}3{\rm{G}}1}$	${\tau }_{\mathrm{lagG}3{\rm{G}}2}$
Name	(s)	(s)	(s)	(s)	(s)	(s)	(s)	(s)	(s)
GRB 970228	−0.456	56.584	−0.002 ± 0.024	53.442 ± 2.891	0.592 ± 0.048	39.280 ± 2.556	⋯	⋯	⋯
GRB 970828	−4.248	94.936	0.864 ± 0.112	66.208 ± 2.781	7.680 ± 0.161	17.792 ± 0.310	⋯	⋯	⋯
GRB 971214	−9.060	16.564	−9.060 ± 2.082	15.892 ± 2.133	−3.172 ± 2.951	6.724 ± 2.970	⋯	⋯	⋯
GRB 990123	−17.312	111.200	1.600 ± 0.161	62.016 ± 1.179	7.904 ± 0.072	26.336 ± 0.757	0.681 ± 0.091	0.619 ± 0.099	0.165 ± 0.050
GRB 990506	−0.390	164.742	1.952 ± 0.041	128.608 ± 0.654	12.032 ± 0.088	83.392 ± 2.565	⋯	⋯	⋯
GRB 990510	−0.320	69.568	0.688 ± 0.186	55.888 ± 8.108	38.976 ± 1.735	5.760 ± 1.745	⋯	⋯	⋯
GRB 990705	−1.698	67.746	1.648 ± 0.066	33.232 ± 1.120	7.488 ± 0.096	14.720 ± 0.211	0.053 ± 0.016	0.103 ± 0.063	0.016 ± 0.014
GRB 990712	−1.637	18.821	−1.637 ± 0.862	16.629 ± 1.777	0.784 ± 0.173	10.784 ± 0.470	⋯	⋯	⋯
GRB 991208	−0.148	76.436	0.688 ± 0.016	63.056 ± 0.481	5.136 ± 1.562	53.680 ± 1.567	⋯	⋯	⋯
GRB 991216	−17.477	44.629	0.672 ± 0.032	14.528 ± 0.140	3.264 ± 0.025	4.704 ± 0.154	⋯	⋯	⋯
GRB 000131	−77.719	105.735	−74.775 ± 2.944	96.471 ± 3.125	−18.839 ± 12.138	27.719 ± 12.280	⋯	⋯	⋯

Note. A positive value of the spectral lag ${\tau }_{\mathrm{lag}}$ corresponds to a delay of the soft photons.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The last three columns of Table 2 present the spectral lags ${\tau }_{\mathrm{lagG}2{\rm{G}}1}$, ${\tau }_{\mathrm{lagG}3{\rm{G}}1}$, and ${\tau }_{\mathrm{lagG}3{\rm{G}}2}$. For the 58 GRBs selected for the spectral lag analysis, the numbers of lags calculated are as follows: ${\tau }_{\mathrm{lagG}2{\rm{G}}1}$ (G2–G1)—55, ${\tau }_{\mathrm{lagG}3{\rm{G}}1}$ (G3–G1)—32, and ${\tau }_{\mathrm{lagG}3{\rm{G}}2}$ (G3–G2)—38. The missing lag values are not constrained; this may be due to a weak detection in one or both analyzed channels, or to a significant difference in a pulse shape between them.

Figure 4 presents the T₅₀, T₉₀, and T₁₀₀ observer- and rest-frame distributions. The rest-frame quantities are the corresponding observer-frame values scaled by the time dilation factor 1/($1+z$).We note that the observer-frame energy band G2+G3, in which the durations are calculated, corresponds to multiple energy bands in the source-frame thus introducing a variable energy-dependant factor which must be accounted for when analyzing the rest-frame durations. The same considerations apply to the spectral lags.

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4.2.1. Analysis

For each burst from our sample, two time intervals were selected for spectral analysis: time-averaged fits were performed over the interval closest to T₁₀₀ (hereafter the TI spectrum); the peak spectrum corresponds to the time when the peak count rate (PCR) is reached. The peak spectrum accumulation time may vary from burst to burst depending on the GRB intensity and the presence of significant spectral evolution. For 38 bursts with poor count statistics, the TI and the peak spectra are measured over the same interval.

More than a dozen bursts from the sample show two or more emission episodes separated by periods of quiescence. In the majority of cases, all emission episodes were included to the TI spectrum. KW triggered on weak precursors of GRB 120716A and GRB 160625B. To maintain a reasonable signal-to-noise ratio, only the main episodes of these bursts contributed to the spectral analysis presented in this paper.

The spectral analysis was performed using XSPEC version 12.9.0 (Arnaud 1996). The raw count rate spectra were rebinned to have a minimum of 20 counts per channel to ensure Gaussian-distributed count statistics and fitted using the ${\chi }^{2}$ statistic. Each spectrum was fitted by two spectral models. The first model is the Band function (hereafter BAND; Band et al. 1993):

$$\begin{eqnarray}f(E)\,\propto \,\left\{\begin{array}{ll}{E}^{\alpha }\exp \left(-\displaystyle \frac{E(2+\alpha )}{{E}_{{\rm{p}}}}\right), & \quad E\lt (\alpha -\beta )\displaystyle \frac{{E}_{{\rm{p}}}}{2+\alpha }\\ {E}^{\beta }{\left[(\alpha -\beta )\displaystyle \frac{{E}_{{\rm{p}}}}{(2+\alpha )}\right]}^{(\alpha -\beta )}\exp \,(\beta -\alpha ), & \quad E\geqslant (\alpha -\beta )\displaystyle \frac{{E}_{{\rm{p}}}}{2+\alpha },\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where α is the low-energy photon index and β is the high-energy photon index. The second spectral model is an exponentially cutoff power-law (CPL), parameterized as E_p:

$$\begin{eqnarray}&&f(E)\propto {E}^{\alpha }\exp \left(-\displaystyle \frac{E(2+\alpha )}{{E}_{{\rm{p}}}}\right).\end{eqnarray} \tag{ 2 }$$

In the only case where both "curved" models result in ill-constrained fits (GRB 080413B), a simple power-law (PL) function was used: $f(E)\propto {E}^{\alpha }$. All the spectral models were normalized to the energy flux (F) in the 10 keV–10 MeV range (observer frame). The fits were performed in the energy range from ∼20 keV to the upper limit of 0.5–15 MeV, depending on the presence of statistically significant GRB emission in the MeV band and, also, on the stability of the background in the upper spectral channels, which are affected, for some GRBs, by solar particles. The parameter errors were estimated using the XSPEC command ERROR based on the change in fit statistic (${\rm{\Delta }}{\chi }^{2}=1$) which corresponds to the 68% CL.

For each spectrum, we present the results for the models whose parameters are constrained (hereafter, GOOD models). The best-fit spectral model (the BEST model) was chosen based on the difference in ${\chi }^{2}$ between the CPL and the BAND fits. The criterion for accepting a model with a single additional parameter is a change in ${\chi }^{2}$ of at least 6 (${\rm{\Delta }}{\chi }^{2}\equiv {\chi }_{\mathrm{CPL}}^{2}-{\chi }_{\mathrm{BAND}}^{2}\gt 6$). This criterion is widely accepted for choosing between nested spectral models in GRB studies (see, e.g., Sakamoto et al. 2008; Krimm et al. 2009; Goldstein et al. 2012) and corresponds to an F test chance improvement probability of ∼0.015 for a reasonably good quality of fit (the reduced chi-squared, i.e., the chi-squared per degree of freedom (d.o.f.), ${\chi }_{{\rm{r}}}^{2}\sim 1$). It should be noted that in the KW GCN circulars a different approach is used for the best-fit model selection: BAND is preferred over CPL in the case of the constrained fit, and not dependent on ${\rm{\Delta }}{\chi }^{2}$.

4.2.2. Results

The 10 columns in Table 3 contain the following information: (1) the GRB name (see Table 1); (2) the spectrum type, where "i" indicates that the spectrum is TI, "p" means that the spectrum is peak; (3) and (4) contain the spectrum start time ${t}_{\mathrm{start}}$ (relative to T₀) and its accumulation time ${\rm{\Delta }}T;$ (5) GOOD models for each spectrum ($\dagger$ indicates the BEST model); (6)–(8) α, β, and ${E}_{{\rm{p}}};$ (9) F (normalization); (10) ${\chi }^{2}/{\rm{d}}.{\rm{o}}.{\rm{f}}.$ along with the null hypothesis probability given in brackets. In cases where the lower limit for β is not constrained, the value of (${\beta }_{\min }-\beta$) is provided instead, where ${\beta }_{\min }=-10$ is the lower limit for the fits. For the best-fit values of $\beta \lt -4$, only the upper limits on β are given.

Table 3.  Spectral Parameters

Burst	Spec.	${t}_{\mathrm{start}}$	${\rm{\Delta }}T$	Model	α	β	${E}_{{\rm{p}}}$	F	${\chi }^{2}/{\rm{d}}.{\rm{o}}.{\rm{f}}.$
Name	Type	(s)	(s)				(keV)	(${10}^{-6}$ erg cm−2 s−1)	(Prob.)
GRB 970228	i	0.000	8.448	CPLa	$-{1.27}_{-0.22}^{+0.24}$	⋯	${165}_{-25}^{+39}$	${0.53}_{-0.05}^{+0.06}$	44.9/56 (0.86)
	i			Band	$-{1.24}_{-0.22}^{+0.31}$	$-{2.82}_{-7.18}^{+0.60}$	${159}_{-36}^{+38}$	${0.60}_{-0.06}^{+0.06}$	44.5/55 (0.84)
	p	0.000	0.256	CPLa	$-{0.81}_{-0.27}^{+0.32}$	⋯	${309}_{-60}^{+102}$	${3.38}_{-0.40}^{+0.42}$	13.3/24 (0.96)
	p			Band	$-{0.76}_{-0.29}^{+0.49}$	$-{2.64}_{-7.36}^{+0.70}$	${286}_{-105}^{+102}$	${4.14}_{-0.47}^{+0.47}$	13.0/23 (0.95)
GRB 970828	i	0.000	70.656	CPL	$-{0.86}_{-0.04}^{+0.04}$	⋯	${346}_{-17}^{+19}$	${0.89}_{-0.03}^{+0.03}$	72.6/66 (0.27)
	i			Banda	$-{0.73}_{-0.06}^{+0.06}$	$-{2.18}_{-0.15}^{+0.11}$	${271}_{-22}^{+24}$	${1.33}_{-0.05}^{+0.05}$	63.5/65 (0.53)
	p	17.920	5.120	CPL	$-{0.91}_{-0.05}^{+0.05}$	⋯	${355}_{-25}^{+29}$	${2.20}_{-0.10}^{+0.10}$	68.6/79 (0.79)
	p			Banda	$-{0.78}_{-0.07}^{+0.08}$	$-{2.18}_{-0.15}^{+0.12}$	${271}_{-28}^{+31}$	${3.23}_{-0.26}^{+0.27}$	57.6/78 (0.96)
GRB 971214	i	0.000	8.448	CPLa	$-{0.50}_{-0.17}^{+0.19}$	⋯	${179}_{-16}^{+20}$	${0.39}_{-0.03}^{+0.03}$	72.2/78 (0.66)
	i			Band	$-{0.32}_{-0.22}^{+0.26}$	$-{2.39}_{-0.35}^{+0.23}$	${154}_{-19}^{+20}$	${0.58}_{-0.09}^{+0.11}$	67.8/77 (0.77)

Note.

^aIndicates the BEST model for the spectrum.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Although KW high-resolution spectra do not cover the pre-trigger emission, for about two-thirds of GRBs in our sample the TI spectra include $\geqslant 90 \%$ of the burst counts (and only for six bursts this fraction is $\lt 50 \%$). A major fraction of the GRB 090812 counts, about one-half of the short GRB 100206A counts, and a significant fraction of the short GRB 070714B counts were accumulated before the trigger. For these bursts, we performed the spectral analysis using both multichannel spectra and the three-channel spectra constructed from light-curve data. Together, these spectra cover the burst T₁₀₀ interval.

Figure 5 shows the distributions of spectral parameters. The CPL model's α for both TI and peak spectra are distributed around $\alpha \approx -1$. For the BAND model, α for the TI and peak spectra are distributed around ≈−1 and ≈−0.85, respectively. The high-energy photon indices β for the TI and peak spectra are distributed around ≈−2.5 and ≈−2.35, respectively. We found BAND to be the BEST model for 54 TI and 51 peak spectra. The remaining spectra (with the exception of GRB 080413B) were best fitted by CPL. E_p for the BEST model varies from $\approx 40\,\mathrm{keV}$ to $\approx 3.5\,\mathrm{MeV}$ (GRB 090510). The TI spectrum ${E}_{{\rm{p}}}$ (${E}_{{\rm{p}},i}$) distributions for both spectral models peak around 250 keV, while the peak spectrum ${E}_{{\rm{p}}}$ (${E}_{{\rm{p}},{\rm{p}}}$) distributions peak around 300 keV. The corresponding rest-frame peak energies, ${E}_{{\rm{p}},i,z}=(1+z){E}_{{\rm{p}},i}$ and ${E}_{{\rm{p}},{\rm{p}},z}\,=(1+z){E}_{{\rm{p}},{\rm{p}}}$, vary from $\approx 50\,\mathrm{keV}$ to $\approx 6.7\,\mathrm{MeV}$ (GRB 090510).

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4.3.1. Fluences and Peak Fluxes

4.3.2. k-correction and Rest-frame Energetics

The cosmological rest-frame energetics, the isotropic-equivalent energy release ${E}_{\mathrm{iso}}$ and the isotropic-equivalent peak luminosity ${L}_{\mathrm{iso}}$, can be calculated, with the proper k-correction, as ${E}_{\mathrm{iso}}=\tfrac{4\pi {D}_{L}^{2}}{1+z}\times S\times k$ and ${L}_{\mathrm{iso}}=4\pi {D}_{L}^{2}\times {F}_{\mathrm{peak}}\times k;$ where ${D}_{L}$ is the luminosity distance. The k-correction to the rest-frame (see, e.g., Bloom et al. 2001b or Kovács et al. 2011 for details) is formulated in terms of spectral model energy flux F as

$$\begin{eqnarray*}&&k=\displaystyle \frac{F[{E}_{1}/(1+z),{E}_{2}/(1+z)]}{F[{e}_{1},{e}_{2}]},\end{eqnarray*}$$

where $[{e}_{1}=10\,\mathrm{keV}$, ${e}_{2}=10$ MeV] is our flux calculation band in the observer frame, and [E₁, E₂] is the rest-frame "bolometric" energy band. For E₁, we accept 1 keV and for E₂, we calculate $(1+z)\cdot {e}_{2}=(1+z)\cdot 10\,\mathrm{MeV}$. The latter value is higher than the widely used rest-frame limit of 10 MeV, since the upper boundary of the KW energy range is rather high ($\gt 10$ MeV) and choosing ${E}_{2}=10\,\mathrm{MeV}$ would narrow the energy band of our observations.

4.3.3. Collimation-corrected Energetics

Knowing ${t}_{\mathrm{jet}}$, one can estimate the collimation-corrected energy released in gamma-rays ${E}_{\gamma }={E}_{\mathrm{iso}}(1-\cos {\theta }_{\mathrm{jet}})$ and the collimation-corrected peak luminosity ${L}_{\gamma }={L}_{\mathrm{iso}}(1-\cos {\theta }_{\mathrm{jet}})$, where ${\theta }_{\mathrm{jet}}$ is the jet opening angle and $(1-\cos {\theta }_{\mathrm{jet}})$ is the collimation factor.

In the case of a CBM with constant number density n, hereafter HM, ${\theta }_{\mathrm{jet}}$ is given by Sari et al. (1999):

$$\begin{eqnarray}&&{\theta }_{\mathrm{jet},\mathrm{HM}}=\displaystyle \frac{1}{6}{\left(\displaystyle \frac{{t}_{\mathrm{jet}}}{1+z}\right)}^{3/8}{\left(\displaystyle \frac{n{\eta }_{\gamma }}{{E}_{\mathrm{iso},52}}\right)}^{1/8},\end{eqnarray} \tag{ 3 }$$

where ${\eta }_{\gamma }$ is the radiative efficiency of the prompt phase, ${E}_{\mathrm{iso},52}$ is the prompt emitted energy in units of 10⁵² erg, and ${t}_{\mathrm{jet}}$ is measured in days. For calculations, we adopted canonical values ${\eta }_{\gamma }=0.2$ and n = 1 cm⁻³ (Frail et al. 2001).

In the case of a stellar-wind-like CBM with $n(r)\propto {r}^{-2}$, hereafter WM, the jet opening angle according to Li & Chevalier (2003) is

$$\begin{eqnarray}&&{\theta }_{\mathrm{jet},\mathrm{WM}}=0.2016{\left(\displaystyle \frac{{t}_{\mathrm{jet}}}{1+z}\right)}^{1/4}{\left(\displaystyle \frac{{\eta }_{\gamma }{A}_{* }}{{E}_{\mathrm{iso},52}}\right)}^{1/4},\end{eqnarray} \tag{ 4 }$$

where ${A}_{* }=({\dot{M}}_{{\rm{w}}}/(4\pi {v}_{{\rm{w}}})$/($5\times {10}^{11}$ g cm⁻¹) is the wind parameter, ${\dot{M}}_{{\rm{w}}}$ is the mass-loss rate due to the wind, and ${v}_{{\rm{w}}}$ is the wind velocity; ${A}_{* }\sim 1$ is typical for a Wolf–Rayet star. Following Ghirlanda et al. (2007), we assume ${A}_{* }=1$ for all bursts with WM neglecting the unknown uncertainty of this parameter.

In this work, we only consider jet breaks that were detected either in optical/IR afterglow light curves or in two spectral bands simultaneously (e.g., in X-ray and in radio). Among ∼60 jet breaks reported for KW GRBs in the literature, 32 meet this criterion (including two for Type I bursts, GRB 051221A and GRB 030603B), and 23 of those GRBs have reasonable constraints on the CBM density profile (14 HM and 9 WM).

4.3.4. Results

Table 4 summarizes observer-frame and non-collimated rest-frame energetics. The first two columns are the GRB name and z. The next seven columns present the observer-frame energetics: S; peak fluxes on the three timescales (${F}_{\mathrm{peak},1024}$ (1024 ms), ${F}_{\mathrm{peak},64}$ (64 ms), and ${F}_{\mathrm{peak},64,{\rm{r}}}$ ($(1+z)\cdot 64$ ms)), together with start times of the intervals when the PCR is reached (${T}_{\mathrm{peak},1024}$, ${T}_{\mathrm{peak},64}$, and ${T}_{\mathrm{peak},64,{\rm{r}}}$). The next two columns contain ${E}_{\mathrm{iso}}$ and the peak isotropic luminosity, ${L}_{\mathrm{iso}}$, calculated from ${F}_{\mathrm{peak},64,{\rm{r}}}$. The provided ${L}_{\mathrm{iso}}$ values may be adjusted to a different timescale ${\rm{\Delta }}T$ (64 or 1024 ms) as:

$$\begin{eqnarray*}&&{L}_{\mathrm{iso}}({\rm{\Delta }}T)=\displaystyle \frac{{F}_{\mathrm{peak}}({\rm{\Delta }}T)}{{F}_{\mathrm{peak},64,{\rm{r}}}}{L}_{\mathrm{iso}}.\end{eqnarray*}$$

Table 4.  Burst Energetics

Burst Name	z	Sb	${T}_{\mathrm{peak},1024}$ c	${F}_{\mathrm{peak},1024}$ d	${T}_{\mathrm{peak},64}$ c	${F}_{\mathrm{peak},64}$ d	${T}_{\mathrm{peak},64,{\rm{r}}}$ c	${F}_{\mathrm{peak},64,{\rm{r}}}$ d	${E}_{\mathrm{iso}}$ e	${L}_{\mathrm{iso}}$ f	${F}_{\mathrm{lim}}$ d	${z}_{\max }$
GRB 970228	0.69	${8.07}_{-0.41}^{+0.49}$	−0.512a	${2.24}_{-0.34}^{+0.43}$	0.106	${4.59}_{-0.78}^{+0.80}$	0.118	${4.12}_{-0.64}^{+0.66}$	12.01 ± 0.93	9.6 ± 1.5	0.92	1.32
GRB 970828	0.96	${101.8}_{-3.6}^{+3.5}$	20.256	${4.30}_{-0.44}^{+0.46}$	20.416	${6.16}_{-0.87}^{+0.88}$	20.416	${6.11}_{-0.71}^{+0.71}$	262.2 ± 9.1	30.9 ± 3.6	1.1	1.85
GRB 971214	3.42	${5.72}_{-0.22}^{+0.24}$	2.752	${0.604}_{-0.068}^{+0.071}$	5.664	${0.99}_{-0.25}^{+0.25}$	−0.280	${0.69}_{-0.11}^{+0.12}$	146.3 ± 6.1	78 ± 13	0.44	4.05
GRB 990123	1.60	${310.6}_{-7.8}^{+8.0}$	5.872	${25.6}_{-1.1}^{+1.1}$	6.048	${29.1}_{-2.3}^{+2.3}$	5.984	${27.7}_{-1.6}^{+1.6}$	2133 ± 54	490 ± 29	3.1	5.04
GRB 990506	1.31	${261.0}_{-8.8}^{+8.8}$	87.040	${9.32}_{-0.46}^{+0.47}$	90.048	${13.8}_{-1.1}^{+1.1}$	87.360	${12.13}_{-0.68}^{+0.69}$	1255 ± 43	134.2 ± 7.6	0.92	3.80
GRB 990510	1.62	${21.72}_{-0.82}^{+0.89}$	44.160	${3.56}_{-0.33}^{+0.35}$	44.224	${5.57}_{-0.78}^{+0.80}$	44.544	${4.35}_{-0.50}^{+0.52}$	174.2 ± 8.1	81.0 ± 9.6	0.76	3.37
GRB 990705	0.84	${109.0}_{-3.8}^{+3.8}$	14.000	${4.80}_{-0.42}^{+0.44}$	15.856	${8.92}_{-0.96}^{+0.96}$	15.824	${8.36}_{-0.76}^{+0.76}$	218.1 ± 7.7	30.7 ± 2.8	0.84	2.02
GRB 990712	0.43	${6.25}_{-0.33}^{+0.38}$	10.880	${0.802}_{-0.087}^{+0.118}$	11.456	${1.10}_{-0.17}^{+0.17}$	11.504	${1.05}_{-0.14}^{+0.14}$	3.86 ± 0.28	1.20 ± 0.18	1.1	0.50
GRB 991208	0.71	${154.9}_{-3.0}^{+2.9}$	56.256	${18.59}_{-0.53}^{+0.55}$	56.960	${22.0}_{-1.1}^{+1.1}$	56.960	${21.88}_{-0.86}^{+0.87}$	233.4 ± 4.6	53.2 ± 2.1	0.61	3.30
GRB 991216	1.02	${297.0}_{-3.7}^{+3.8}$	3.840	${47.6}_{-3.1}^{+3.2}$	4.032	${99.4}_{-4.7}^{+4.7}$	3.952	${86.9}_{-3.6}^{+3.6}$	886 ± 11	510 ± 21	1.6	5.73
GRB 000131	4.50	${45.3}_{-1.3}^{+1.3}$	2.144	${3.09}_{-0.26}^{+0.27}$	2.880	${5.68}_{-0.63}^{+0.63}$	2.672	${3.99}_{-0.28}^{+0.28}$	1817 ± 56	859 ± 60	0.67	8.48

Notes.

^aSince the trigger mode light curve begins at ${T}_{0}-0.512\,{\rm{s}}$, the corresponding peak energy flux may be underestimated due to the absence of high-res data before ${T}_{0}-0.512\,{\rm{s}}$. ^bIn units of ${10}^{-6}$ erg cm⁻². ^cThe start time of the time interval when the peak count rate is reached, s. ^dIn units of ${10}^{-6}$ erg cm⁻² s⁻¹. ^eIn units of 10⁵¹ erg. ^fIn units of 10⁵¹ erg s⁻¹.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The rightmost columns provide two additional characteristics useful when the sample selection effects are considered: the bolometric energy flux corresponding to the GRB detection threshold, ${F}_{\mathrm{lim}}$ (Section 5.3); and ${z}_{\max }$, the GRB detection horizon described in Section 5.4.

In Figure 6, the distributions of S, ${F}_{\mathrm{peak},64}$, ${E}_{\mathrm{iso}}$, and ${L}_{\mathrm{iso}}$ are shown. The most fluent burst in this catalog is GRB 130427A ($S=2.86\times {10}^{-3}$ erg cm⁻²). The brightest burst based on the peak energy flux is GRB 110918A (${F}_{\mathrm{peak},64}=9.02\,\times {10}^{-4}$ erg cm⁻² s⁻¹). The most energetic burst in terms of the isotropic energy is GRB 090323 (${E}_{\mathrm{iso}}=5.81\times {10}^{54}$ erg). The most luminous burst contained in this catalog is GRB 110918A (${L}_{\mathrm{iso}}=4.65\times {10}^{54}$ erg s⁻¹).

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Table 5 summarizes the collimation-corrected energetics for 32 bursts with "reliable" jet break times. The first column is the burst name. The next three columns specify ${t}_{\mathrm{jet}}$, the CBM environment implied (HM or WM), and references to them. The next two columns contain the derived ${\theta }_{\mathrm{jet}}$ and the corresponding collimation factor, and the last two columns present E_γ and L_γ. For bursts with no reasonable constraint on the CBM profile the results are given for both HM and WM.

Table 5.  Collimation-corrected Parameters

Burst	${t}_{\mathrm{jet}}$	CBM	Referencesa	${\theta }_{\mathrm{jet}}$	Collimation	${E}_{\gamma }$	${L}_{\gamma }$
Name	(days)			(deg)	factor ($\times {10}^{-3}$)	(1049 erg)	(1049 erg s−1)
GRB 990123	${2.06}_{-0.83}^{+0.83}$	WM	(1)	1.91 ± 0.19	0.55 ± 0.12	118.10 ± 24.10	27.16 ± 5.74
GRB 990510	${1.31}_{-0.07}^{+0.07}$	HM	(1)	4.21 ± 0.09	2.70 ± 0.11	47.07 ± 3.20	21.90 ± 0.95
GRB 990705	1b	HMc	(2)	4.23 ± 0.32	2.72 ± 0.42	59.31 ± 9.13	8.36 ± 1.26
		WM		3.07 ± 0.16	1.43 ± 0.15	31.28 ± 3.21	4.41 ± 0.41
GRB 990712	${1.61}_{-0.19}^{+0.19}$	HMc	(3)	9.20 ± 0.42	12.90 ± 1.19	4.96 ± 0.68	1.54 ± 0.28
		WM		10.10 ± 0.35	15.50 ± 1.09	5.98 ± 0.60	1.85 ± 0.34
GRB 991216	${1.1}_{-0.13}^{+0.13}$	WM	(1)	2.16 ± 0.06	0.71 ± 0.04	63.13 ± 3.88	36.32 ± 2.68
GRB 000301C	${7.3}_{-0.5}^{+0.5}$	HM	(4)	9.33 ± 0.30	13.20 ± 0.85	44.56 ± 7.20	74.95 ± 8.08
GRB 000418	${7.85}_{-2.71}^{+2.71}$	HMc	(1)	9.62 ± 1.25	14.10 ± 3.87	134.50 ± 44.78	53.39 ± 16.34
		WM		6.09 ± 0.53	5.65 ± 1.03	54.04 ± 12.13	21.44 ± 4.70
GRB 000926	${2.1}_{-0.15}^{+0.15}$	WM	(1)	3.07 ± 0.06	1.43 ± 0.06	39.83 ± 2.46	16.24 ± 2.05
GRB 010222	${0.93}_{-0.06}^{+0.15}$	WM	(5), (1)	1.88 ± 0.06	0.54 ± 0.03	57.63 ± 3.81	12.56 ± 1.17
GRB 010921	${35}_{-5}^{+5}$	HM	(6)	25.51 ± 1.37	97.50 ± 10.60	105.70 ± 16.10	16.98 ± 2.43
GRB 011121	${1.54}_{-0.22}^{+0.22}$	WM	(1)	4.49 ± 0.16	3.07 ± 0.23	30.39 ± 2.26	4.09 ± 0.42
GRB 020405	${2.4}_{-0.45}^{+0.45}$	WM	(1)	4.56 ± 0.21	3.16 ± 0.30	37.07 ± 3.65	5.45 ± 0.74
GRB 020813	${0.77}_{-0.25}^{+0.25}$	HM	(1)	3.04 ± 0.37	1.41 ± 0.36	106.70 ± 27.01	22.22 ± 5.49
GRB 030329	${0.69}_{-0.06}^{+0.08}$	HM	(7)	6.02 ± 0.23	5.51 ± 0.43	9.11 ± 0.87	1.23 ± 0.10
GRB 041006	${0.23}_{-0.04}^{+0.04}$	WM	(1)	5.13 ± 0.23	4.00 ± 0.37	2.75 ± 0.26	2.15 ± 0.42
GRB 050401	${1.5}_{-0.5}^{+0.5}$	HMc	(8)	3.38 ± 0.42	1.74 ± 0.46	80.59 ± 21.26	36.98 ± 10.05
		WM		2.33 ± 0.20	0.83 ± 0.14	38.38 ± 6.78	17.61 ± 3.45
GRB 050525A	${0.152}_{-0.008}^{+0.008}$	HMc	(9)	2.83 ± 0.06	1.22 ± 0.05	3.43 ± 0.17	2.33 ± 0.10
		WM		3.31 ± 0.05	1.67 ± 0.05	4.68 ± 0.16	3.18 ± 0.19
GRB 050820A	${18}_{-2}^{+2}$	HM	(10)	7.99 ± 0.33	9.70 ± 0.83	1005.00 ± 95.19	133.80 ± 12.13
GRB 051221A	5b	HM	(11)	14.04 ± 1.06	29.90 ± 4.66	9.20 ± 1.51	67.56 ± 10.47
GRB 060614	${1.31}_{-0.03}^{+0.03}$	HM	(12)	9.72 ± 0.11	14.30 ± 0.32	3.89 ± 0.31	0.42 ± 0.02
GRB 061121	1.16b	HM	(13)	3.94 ± 0.30	2.36 ± 0.37	71.67 ± 11.46	53.35 ± 8.32
GRB 070125	3.78b	HM	(14)	4.94 ± 0.37	3.71 ± 0.58	474.00 ± 74.85	108.20 ± 16.58
GRB 071010B	${3.44}_{-0.39}^{+0.39}$	HMc	(15)	9.22 ± 0.41	12.90 ± 1.16	18.72 ± 2.71	8.55 ± 1.06
		WM		8.12 ± 0.30	10.00 ± 0.74	14.50 ± 1.61	6.62 ± 1.22
GRB 080319B	${11.6}_{-1}^{+1}$	WM	(16), (12)	3.41 ± 0.07	1.77 ± 0.08	278.10 ± 12.20	21.00 ± 1.30
GRB 090328	${9}_{-6}^{+11.6}$	HM	(17), (12)	10.73 ± 4.13	17.50 ± 16.00	190.20 ± 149.00	59.56 ± 46.09
GRB 090618	${0.5}_{-0.11}^{+0.11}$	HM	(18)	3.42 ± 0.28	1.78 ± 0.31	45.04 ± 8.08	4.74 ± 0.79
GRB 090926A	${10}_{-2}^{+2}$	HM	(17), (12)	6.20 ± 0.47	5.85 ± 0.91	1234.00 ± 188.70	549.80 ± 82.86
GRB 091127	${0.39}_{-0.02}^{+0.02}$	HMc	(19)	4.46 ± 0.09	3.02 ± 0.13	4.82 ± 0.63	3.45 ± 0.29
		WM		4.92 ± 0.10	3.68 ± 0.15	5.87 ± 0.63	4.20 ± 0.74
GRB 110503A	${1.06}_{-0.14}^{+0.14}$	HMc	(20)	3.80 ± 0.19	2.20 ± 0.23	46.80 ± 5.07	42.85 ± 4.31
		WM		2.87 ± 0.10	1.26 ± 0.09	26.71 ± 1.93	24.46 ± 2.28
GRB 130427A	${0.43}_{-0.05}^{+0.05}$	HM	(21)	2.91 ± 0.13	1.29 ± 0.12	114.80 ± 10.09	35.64 ± 3.11
GRB 130603B	${0.47}_{-0.06}^{+0.02}$	HMc	(22)	6.43 ± 0.23	6.29 ± 0.46	1.23 ± 0.10	18.79 ± 1.37
		WM		8.90 ± 0.24	12.00 ± 0.66	2.36 ± 0.13	36.00 ± 3.00
GRB 151027A	2.3b	WM	(23)	6.08 ± 0.36	5.63 ± 0.68	18.60 ± 2.74	4.41 ± 0.73

Notes.

^aIn cases where two references are given, the first one corresponds to the ${t}_{\mathrm{jet}}$ estimate and the second one corresponds to the preferred CBM. ^bWhen no ${t}_{\mathrm{jet}}$ uncertainty is available from the literature, we take the sample-mean $\sim 0.2\cdot {t}_{\mathrm{jet}}$ as a 68% ${t}_{\mathrm{jet}}$ error for the calculations. ^cIn cases where no preferred CBM density profile is available from the literature, we provide the estimates for both HM and WM.

References. (1) Zeh et al. (2006), (2) Masetti et al. (2000), (3) Björnsson et al. (2001), (4) Berger et al. (2000), (5) Galama et al. (2003), (6) Price et al. (2003), (7) Resmi et al. (2005), (8) Ghirlanda et al. (2007), (9) Blustin et al. (2006), (10) Cenko et al. (2006), (11) Soderberg et al. (2006), (12) Schulze et al. (2011), (13) Page et al. (2007), (14) Chandra et al. (2008), (15) Kann et al. (2007), (16) Tanvir et al. (2010a), (17) Cenko et al. (2011), (18) Cano et al. (2011), (19) Filgas et al. (2011), (20) Kann et al. (2011), (21) Maselli et al. (2014), (22) Fong et al. (2014), (23) Nappo et al. (2017).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 6.  Parameter Statistics

Parameter	Min	Max	Mean	Median
Name	Value	Value	Value	Value
Redshift	0.096	5	1.50	1.32
T100 (s)	0.124	484.858	67.689	37.312
T90 (s)	0.070	440.826	46.557	21.664
T50 (s)	0.034	167.290	16.959	7.616
${T}_{100,z}$ (s)	0.088	170.884	29.476	13.974
${T}_{90,z}$ (s)	0.052	121.954	19.447	9.677
${T}_{50,z}$ (s)	0.025	49.733	7.220	3.002
${\tau }_{\mathrm{lagG}2{\rm{G}}1}$ (ms)	0.6	2495	292	150
${\tau }_{\mathrm{lagG}3{\rm{G}}1}$ (ms)	4.8	5106	543	343
${\tau }_{\mathrm{lagG}3{\rm{G}}2}$ (ms)	2.1	765	176	132
${\tau }_{\mathrm{lagG}2{\rm{G}}1,z}$ (ms)	0.4	1290	143	68
${\tau }_{\mathrm{lagG}3{\rm{G}}1,z}$ (ms)	3.7	2630	257	133
${\tau }_{\mathrm{lagG}3{\rm{G}}2,z}$ (ms)	1.4	388	85	68
${E}_{{\rm{p}},i}$ (keV), Type I GRBs	468	3516	953	640
${E}_{{\rm{p}},{\rm{p}}}$ (keV) Type I GRBs	468	3386	966	671
${E}_{{\rm{p}},i,z}$ (keV) Type I GRBs	658	6691	1637	988
${E}_{{\rm{p}},{\rm{p}},z}$ (keV) Type I GRBs	658	6444	1647	991
${E}_{{\rm{p}},i}$ (keV) Type II GRBs	37	1083	298	238
${E}_{{\rm{p}},{\rm{p}}}$ (keV) Type II GRBs	37	1511	360	271
${E}_{{\rm{p}},i,z}$ (keV) Type II GRBs	54	2703	775	661
${E}_{{\rm{p}},{\rm{p}},z}$ (keV) Type II GRBs	53	5137	931	752
S (erg cm−2)	1.13 × 10−6	2.86 × 10−3	1.07 × 10−4	2.51 × 10−5
Fpeak,1024 (erg cm−2 s−1)	5.56 × 10−7	5.08 × 10−4	1.42 × 10−5	3.45 × 10−6
${F}_{\mathrm{peak},64}$ (erg cm−2 s−1)	9.51 × 10−7	9.02 × 10−4	2.55 × 10−5	6.19 × 10−6
${F}_{\mathrm{peak},64,{\rm{r}}}$ (erg cm−2 s−1)	6.89 × 10−7	8.71 × 10−4	2.33 × 10−5	5.41 × 10−6
${E}_{\mathrm{iso}}$ (erg)	4.18 × 1049	5.81 × 1054	5.55 × 1053	1.93 × 1053
${L}_{\mathrm{iso}}$ (erg s−1)	2.94 × 1050	4.65 × 1054	2.55 × 1053	8.32 × 1052
Collimation factor	5.4 × 10−4	3.0 × 10−2	6.5 × 10−3	3.2 × 10−3
${E}_{\gamma }$ (erg)	1.70 × 1049	1.23 × 1052	1.04 × 1051	3.98 × 1050
${L}_{\gamma }$ (erg s−1)	4.22 × 1048	5.50 × 1051	4.39 × 1050	1.62 × 1050

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The jet opening angles vary from 19 to 255 and the corresponding collimation factors from $5.5\times {10}^{-4}$ to 0.098. The brightest KW GRB in terms of both ${E}_{\gamma }$ and ${L}_{\gamma }$ is GRB 090926A (${E}_{\gamma }\simeq 1.23\times {10}^{52}$ erg, ${L}_{\gamma }\simeq 5.50\times {10}^{51}$ erg s⁻¹, ${\theta }_{\mathrm{jet}}\,\simeq 6\buildrel{\circ}\over{.} 20$). The distributions of ${E}_{\gamma }$ and ${L}_{\gamma }$ are shown in Figure 6 and Table 6 provides descriptive statistics of the GRB parameters estimated in Section 4.

Preliminary results on the KW detections of bursts with known redshifts have been reported in more than 100 GCN circulars and the more detailed KW GRB analyses were presented in multiple refereed publications. Although the latter results are, with a few exceptions, statistically consistent with those reported here, the main advantage of this catalog, in comparison to the previous work, is in the use of the unified, systematic approach to re-analyse all 150 bursts in the sample. Particularly, GRB durations were calculated in the G2+G3 band that is less affected by the hard X-ray background variations; this also allows one to separate the hard GRB prompt emission from the emerging X-ray afterglow. The spectral analysis presented here gains an advantage from the most recent and accurate KW DRM; it also relies on a standard procedure for the TI spectrum interval selection based on T₁₀₀. The burst energetics, S and ${F}_{\mathrm{peak}}$ are estimated, in this work, based on the BEST spectral models for TI and peak spectra, which also improves the flux calculation uncertainties. Finally, the reported rest-frame energetics rely on the k-correction procedure that utilizes the full spectral band of the instrument, and they are estimated using a common set of cosmology parameters. To summarize, the results presented in this catalog form a consistent set of observer- and rest-frame GRB parameters useful for further systematic studies.

5.2.1. The Sample Statistics and Comparison of KW with BATSE and GBM Bursts

5.2.2. Spectral Indices

Selection effects are distortions or biases that usually occur when the observational sample is not representative of the true, underlying population. They play a crucial role for GRBs (Turpin et al. 2016; Dainotti & Del Vecchio 2017), which are particularly affected by the Malmquist bias effect that favors the brightest objects against faint objects at large distances, and these biases have to be taken into account when using GRBs as distance estimators, cosmological probes, and model discriminators.

For the sample of the KW triggered-mode GRBs with known redshifts, the selection effects fall into two categories: the KW-specific effects, caused by its trigger sensitivity to the burst prompt emission parameters; and the "external" biases arising in the process of localization and securing GRB redshifts, which are outside of the scope of this paper.

The KW triggered mode is activated when the count rate in the G2 window exceeds a $\approx 9\sigma$ threshold above the background on one of two fixed timescales ${\rm{\Delta }}{T}_{\mathrm{trig}}$, 1 s (applicable, with a few exceptions, to Type II bursts in our sample) or 140 ms (the Type I bursts). Thus, the burst's detection significance may be characterized by a PCR to background statistical uncertainty ratio (over the corresponding ${\rm{\Delta }}{T}_{\mathrm{trig}}$). Although the KW trigger criterion cannot be easily translated into the GRB prompt emission characteristics (e.g., duration, rise-time, spectral shape, or energy flux), an investigation into how their combination may affect the trigger sensitivity to a specific burst may be done indirectly.

We estimated the energy flux sensitivity of the KW detectors following the methodology described in Band (2003). Figure 7 presents the limiting energy flux (10 keV–10 MeV) as a function of ${E}_{{\rm{p}}}$ for ${\rm{\Delta }}{T}_{\mathrm{trig}}=1$ s, for a burst incident angle $60^\circ$, and the S1 detector calibration as of mid-2015. As can be seen, the energy flux threshold under these assumptions is $\approx 1\times {10}^{-6}$ erg cm⁻² s⁻¹ and there is a bias against the detection of soft-spectrum bursts with ${E}_{{\rm{p}}}\lesssim 70\mbox{--}80\,\mathrm{keV}$, especially with CPL spectra, due to the instrumental selection. Meanwhile, the F − E_p diagram stresses the lack of bright ($F\gtrsim 5\times {10}^{-6}$ erg cm⁻² s⁻¹) and soft (${E}_{{\rm{p}}}\lesssim 100$ keV) GRBs, that should be easily detectable with KW. Since the lower boundary of this region is defined by GRBs with moderate-to-high detection significance, the instrumental biases do not affect the sample from this edge of the distribution. Thus, the apparent lack of soft/bright burst observations in the KW sample is likely due to an intrinsic GRB property (see Section 5.7 for more discussion).

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In a similar way, we calculated a limiting observer-frame energy flux for each GRB from the sample using its BEST-model spectral parameters, incident angle, and detector calibration. In order to make the results more helpful for the rest-frame energy calculations, we applied k-corrections (Section 4.3) to these values using the burst redshift. The resulting bolometric limiting fluxes, ${F}_{\mathrm{lim}}$, are given in Table 4; the sample mean value of ${F}_{\mathrm{lim}}$ for the Type II GRBs is $1.08\times {10}^{-6}$ erg cm⁻² s⁻¹. We note that ${F}_{\mathrm{lim}}$ are calculated using the 1 s scale; when compared to peak fluxes determined on a different ${\rm{\Delta }}T$ they should be adjusted as:

$$\begin{eqnarray*}&&{F}_{\mathrm{lim}}({\rm{\Delta }}T)=\displaystyle \frac{{F}_{\mathrm{peak}}({\rm{\Delta }}T)}{{F}_{\mathrm{peak},1024}}{F}_{\mathrm{lim}}.\end{eqnarray*}$$

Figure 8 shows the KW GRB distributions in the ${E}_{\mathrm{iso}}$–z, ${L}_{\mathrm{iso}}$–z, and ${E}_{{\rm{p}},z}$–z diagrams. The region in the ${L}_{\mathrm{iso}}$–z plane above the limit corresponding to ${F}_{\mathrm{lim}}\sim 1\,\times {10}^{-6}$ erg cm⁻² s⁻¹ may be considered free from the selection bias. In the ${E}_{\mathrm{iso}}$–z plane, the selection-free region lies above the limit, corresponding to the bolometric fluence ${S}_{\mathrm{lim}}\sim 3\,\times {10}^{-6}$ erg cm⁻². As mentioned above, the KW detector sensitivity drops rapidly as E_p approaches the lower boundary of the instrument's band (∼20–25 keV as of 2015), and this results in a lack of bursts below ${E}_{{\rm{p}},z,\mathrm{lim}}\approx {(1+z)}^{2}\cdot 25\,\mathrm{keV}$ in the ${E}_{{\rm{p}},z}$–z plane; the additional factor $(1+z)$ here is due to cosmological time dilation.

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Finally, our sample does not exhibit any direct selection effects due to GRB duration. However, some bursts with very gradual rising slopes may not trigger the instrument despite being bright enough to do it in other circumstances. We estimate the number of such GRBs with known redshifts to be $\lesssim 5$ (as of mid-2016); these bursts will be considered, along with other KW background-mode GRBs with known redshifts, in the second part of the catalog.

Knowing the maximum distance at which a particular GRB can be detected by the instrument (the GRB "horizon," ${z}_{\max }$) may be useful in a number of applications, e.g., for deriving the $V/{V}_{\max }$ statistic (Schmidt et al. 1988) or for accounting for the instrumental bias when studying the "true" GRB energy distribution (Atteia et al. 2017).

A common approach to estimate the GRB horizon is to find a redshift ${z}_{\max ,{\rm{L}}}$, at which the limiting isotropic luminosity ${L}_{\mathrm{iso},\mathrm{lim}}=4\pi \ {D}_{L}^{2}\times {F}_{\mathrm{lim}}$, defined by the limiting energy flux estimated for the whole sample (the "monolithic" ${F}_{\mathrm{lim}}$), starts to exceed the GRB ${L}_{\mathrm{iso}}$. The KW trigger, however, is based on a simple photon-counting algorithm and not directly sensitive to the incident energy flux. Thus, the correctness of the described approach (hereafter the monolithic ${F}_{\mathrm{lim}}$ method), which doesn't account for the burst-specific instrumental issues, such as trigger sensitivity to the GRB incident angle, its light-curve shape, and the shape of the energy spectrum, needs an additional confirmation.

When evaluating how GRB detectability by KW changes when the burst source is shifted from its redshift z to a more distant $z^{\prime}$, at least three effects have to be accounted for. First, the solid angle factor, which reduces (assuming identical beaming) an incident bolometric photon flux P by ${({D}_{M}(z)/{D}_{M}(z^{\prime} ))}^{2}$, where D_M is the transverse comoving distance. Second, the cosmological time dilation, which results in the light curve broadening and an additional decrease in P by a factor of (1 + $z^{\prime}$)/(1 + z). Finally, the spectral cutoff, which is inherent to GRB spectra, is redshifted by the same factor, thus decreasing the fraction of P within the instrument trigger band (G2). We estimate the KW detection horizon as a redshift $z^{\prime} ={z}_{\max }$, at which the PCR in the G2 light curve drops below the trigger threshold ($9\sigma$) on both KW trigger scales ${\rm{\Delta }}{T}_{\mathrm{trig}}$ (140 ms and 1 s). ${\mathrm{PCR}}_{z^{\prime} }({\rm{\Delta }}{T}_{\mathrm{trig}})$ is calculated as

$$\begin{eqnarray}{\mathrm{PCR}}_{z^{\prime} }({\rm{\Delta }}{T}_{\mathrm{trig}}) & = & a\times {\mathrm{PCR}}_{z}(a\cdot {\rm{\Delta }}{T}_{\mathrm{trig}})\\ & & \times \displaystyle \frac{{N}_{{\rm{G}}2}(\alpha ,\beta ,a\cdot {E}_{{\rm{p}},{\rm{p}}})}{{N}_{{\rm{G}}2}(\alpha ,\beta ,{E}_{{\rm{p}},{\rm{p}}})}\times {\left(\displaystyle \frac{{D}_{{\rm{M}}}(z)}{{D}_{{\rm{M}}}(z^{\prime} )}\right)}^{2},\end{eqnarray} \tag{ 5 }$$

where $a=(1+z)/(1+z^{\prime} );$ ${\mathrm{PCR}}_{z}(a\cdot {\rm{\Delta }}{T}_{\mathrm{trig}})$ is the PCR reached in the observed G2 light curve on the modified timescale; ${N}_{{\rm{G}}2}(\alpha ,\beta ,{E}_{{\rm{p}},{\rm{p}}})$ is the BEST spectral model count flux in G2 calculated using the DRM; and ${N}_{{\rm{G}}2}(\alpha ,\beta ,a\cdot {E}_{{\rm{p}},{\rm{p}}})$ is the corresponding flux in the redshifted spectrum. The resulting values of ${z}_{\max }$ are given in Table 4 and shown in Figure 9. We found that for both Type I and Type II GRBs, ${z}_{\max }$ are distributed narrowly around the corresponding ${z}_{\max ,{\rm{L}}}$ values calculated assuming the bolometric ${F}_{\mathrm{lim}}=1\times {10}^{-6}$ erg cm⁻² s⁻¹ with the mean and σ of the $(1+{z}_{\max })/(1+{z}_{\max ,{\rm{L}}})$ distribution of 1.01 and 0.12, respectively. Although in some cases ${z}_{\max ,{\rm{L}}}$ calculated with the simple monolithic ${F}_{\mathrm{lim}}$ method may differ from the more precisely evaluated ${z}_{\max }$ by a factor of ∼1.5, our calculations support the general correctness of the former approach.

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The most distant GRB horizon for the KW sample (${z}_{\max }\approx 16.6$) is reached for the ultra-luminous GRB 110918A⁸ at $z=0.981$ and the second-highest (${z}_{\max }\approx 12.5$) is for GRB 050603 ($z=2.82$). At $z\approx 16.6$, the age of the universe amounts to only ∼230 Myr, i.e., a burst that occurred close to the end of the cosmic Dark Ages could still trigger the KW detectors, and a thorough temporal and spectral analysis in a wide observer-frame energy range could be performed. Among the KW Type I GRBs, the highest ${z}_{\max }\approx 5.3$ is for GRB 160410A ($z=1.72$).

Among various statistical parameters, the luminosity function as well as the cosmic formation rate of GRBs are particularly interesting. The luminosity function (LF) is a measure of the number of bursts per unit luminosity, that provides information on the energy release and emission mechanism of GRBs. The cosmic GRB formation rate (GRBFR) is a measure of the number of events per comoving volume and time, which can help us to understand the production of GRBs at various stages of the universe. While LF was originally used to study long-lasting and relatively stable astrophysical phenomena, such as stars and galaxies, the isotropic energy release function (EF, the number of bursts per unit ${E}_{\mathrm{iso}}$) can be more representative for transient phenomena, e.g., for GRBs. The GRB EF was constructed for the first time by Wu et al. (2012) using a sample of 95 bursts with measured redshifts. The KW sample presented in this catalog provides an excellent opportunity to test GRB LF, EF, and GRBFR on an independent data set.

Without loss of generality, the total LF ${\rm{\Phi }}({L}_{\mathrm{iso}},z)$ ⁹ can be rewritten as ${\rm{\Phi }}({L}_{\mathrm{iso}},z)=\rho (z)\phi ({L}_{\mathrm{iso}}/g(z),{\alpha }_{s})/g(z),$ where $\rho (z)$ is the GRB formation rate (GRBFR), $\phi ({L}_{\mathrm{iso}}/g(z))$ is the local LF, $g(z)$ is the luminosity evolution that parameterizes the correlation between L and z, and ${\alpha }_{s}$ is the shape of the LF, whose effect is commonly ignored as the shape of the LF does not change significantly with z (e.g., Yonetoku et al. 2004). Following Lloyd-Ronning et al. (2002), Yonetoku et al. (2004), Wu et al. (2012), and Yu et al. (2015) we chose the functional form of $g{(z)=(1+z)}^{\delta }$ for the luminosity evolution. It should be noted that the isotropic luminosity evolution can be determined by either the evolution of the amount of energy per unit time emitted by the GRB progenitor or by the jet opening angle evolution (see, e.g., Lloyd-Ronning et al. (2002) for the discussion); we tested the KW sample for a correlation between the collimation factor and z and found the correlation negligible (the Spearman rank-order correlation coefficient ${\rho }_{S}=-0.26$ (the corresponding p value ${P}_{{\rho }_{S}}=0.17$) for the subsample of 30 Type II bursts with known collimation factors).

The KW z–${L}_{\mathrm{iso}}$ and z–${E}_{\mathrm{iso}}$ samples suffer from selection effects due to the detection limit of the instrument (see Section 5.3 for details) that results in data truncation seen in Figure 8. To estimate LF, EF, and GRBFR for the sample of 137 KW Type II bursts we used the non-parametric Lynden-Bell ${C}^{-}$ technique (Lynden-Bell 1971) further advanced by Efron & Petrosian (1992) (the EP method); the details of our calculations are described in the Appendix. The EP method is specifically designed to reconstruct the intrinsic distributions from the observed distributions, which accounts for the data truncations introduced by observational bias and includes the effects of the possible correlation between the two variables.

Applying the EP technique based on the individual (i.e., calculated for each burst independently) truncation limits to the z–${L}_{\mathrm{iso}}$ plane, we found that the independence of the variables is rejected at ${\tau }_{0}\equiv \tau (\delta =0)\sim 1.7$ (where τ is a modified version of the Kendall statistic, see the Appendix), and the best luminosity evolution index is ${\delta }_{L}={1.7}_{-0.9}^{+0.9}$ ($1\sigma$ CL). Similar results were obtained using the "monolithic" truncation limit ${F}_{\mathrm{lim}}=2\,\times {10}^{-6}$ erg cm⁻² s⁻¹: ${\tau }_{0}\sim 1.2$ and ${\delta }_{L}={1.7}_{-1.1}^{+0.9}$.

Applying the same method to the z–${E}_{\mathrm{iso}}$ plane and using the monolithic truncating fluence ${S}_{\mathrm{lim}}=4.3\times {10}^{-6}$ erg cm⁻² (see the Appendix for the details of ${F}_{\mathrm{lim}}$ and ${S}_{\mathrm{lim}}$ selection), we found that the independence of the variables is rejected at $\sim 1.6\sigma$, and the best isotropic energy evolution index is ${\delta }_{E}={1.1}_{-0.7}^{+1.5}$. Thus, the estimated ${E}_{\mathrm{iso}}$ and ${L}_{\mathrm{iso}}$ evolutions are comparable. The evolution PL indices ${\delta }_{L}$ and ${\delta }_{E}$ derived here are shallower than those reported in the previous studies: ${\delta }_{L}={2.60}_{-0.20}^{+0.15}$ (Yonetoku et al. 2004), ${\delta }_{L}={2.30}_{-0.51}^{+0.56}$ (Wu et al. 2012), ${\delta }_{L}={2.43}_{-0.38}^{+0.41}$ (Yu et al. 2015), and ${\delta }_{E}={1.80}_{-0.63}^{+0.36}$ (Wu et al. 2012), albeit within errors.

After eliminating the luminosity and energy release evolution, we, following Lynden-Bell (1971), obtained the local cumulative LF and EF, $\psi (L^{\prime} )$ and $\psi (E^{\prime} )$, where $L^{\prime} ={L}_{\mathrm{iso}}/{(1+z)}^{{\delta }_{L}}$ and $E^{\prime} ={E}_{\mathrm{iso}}/{(1+z)}^{{\delta }_{E}}$. We approximated the variance of $\psi (L^{\prime} )$ and $\psi (E^{\prime} )$ by bootstrapping the initial sample and fitted the distributions with a broken power-law (BPL) function:

$$\begin{eqnarray*}\psi (x)\propto \left\{\begin{array}{ll}{x}^{{\alpha }_{1}}, & \quad x\leqslant {x}_{b},\\ {x}_{b}^{({\alpha }_{1}-{\alpha }_{2})}{x}^{{\alpha }_{2}}, & \quad x\gt {x}_{b},\end{array}\right.\end{eqnarray*}$$

where ${\alpha }_{1}$ and ${\alpha }_{2}$ are the PL indices at the dim and bright distribution segments and x_b is the breakpoint of the distribution, and with the CPL function¹⁰ : $\psi (x)\propto {x}^{\alpha }\ \exp (-x/{x}_{\mathrm{cut}})$, where ${x}_{\mathrm{cut}}$ is the cutoff luminosity (or energy).

The fits were performed in $\mathrm{log}-\mathrm{log}$ space using ${\chi }^{2}$ minimization, the results are given in Table 7 and shown in Figure 10 (right panel). The derived BPL slopes of LF and EF are close to each other, both for the dim and bright segments, thus the shape of EF is similar to that of LF; also, these indices are roughly consistent with the LF and EF slopes obtained in Yonetoku et al. (2004) and Wu et al. (2012). The small reduced ${\chi }^{2}$ obtained for both models do not allow us to reject any of them; however, when compared to BPL, the CPL fit to $\psi (L^{\prime} )$ results in a considerably worse quality (${\chi }_{\mathrm{CPL}}^{2}-{\chi }_{\mathrm{BPL}}^{2}\gt 16$); with the PL slope $\alpha \sim -0.6$ and the cutoff luminosity $L{{\prime} }_{\mathrm{cut}}\sim 2.3\times {10}^{54}$ erg s⁻¹. Conversely, the cutoff PL fits $\psi (E^{\prime} )$ better when compared to BPL (${\chi }_{\mathrm{CPL}}^{2}-{\chi }_{\mathrm{BPL}}^{2}\sim -5.5$); with $\alpha \sim -0.35$ and the cutoff energy $E{{\prime} }_{\mathrm{cut}}\sim 2.3\times {10}^{54}$ erg. The existence of a sharp cutoff of the isotropic energy distribution distribution of KW and Fermi-GBM GRBs around $\sim (1\mbox{--}3)\times {10}^{54}$ erg was suggested recently by Atteia et al. (2017).

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Table 7.  LF and EF Fits with BPL and Cutoff PL

Data	Evolution	Model	${\chi }^{2}({\rm{d}}.{\rm{o}}.{\rm{f}}.)$	${\alpha }_{1}$	${\alpha }_{2}$	$\mathrm{log}$ xb,52
	(PL Index)					($\mathrm{log}$ ${x}_{\mathrm{cut},52}$)
$\psi (L^{\prime} )$	${\delta }_{L}=1.7$	BPL	2.05 (133)	−0.47 ± 0.06	−1.05 ± 0.11	0.27 ± 0.12
$\psi (L^{\prime} )$	${\delta }_{L}=1.7$	CPL	18.5 (134)	−0.60 ± 0.04	⋯	2.10 ± 0.15
$\psi (E^{\prime} )$	${\delta }_{E}=1.1$	BPL	19.2 (126)	−0.36 ± 0.01	−1.28 ± 0.11	1.30 ± 0.04
$\psi (E^{\prime} )$	${\delta }_{E}=1.1$	CPL	12.7 (127)	−0.31 ± 0.02	⋯	2.09 ± 0.04
$\psi ({L}_{\mathrm{iso}})$	no evolution	BPL	2.32 (133)	−0.48 ± 0.06	−1.00 ± 0.10	0.96 ± 0.15
$\psi ({L}_{\mathrm{iso}})$	no evolution	CPL	8.90 (134)	−0.54 ± 0.04	⋯	2.58 ± 0.11
$\psi ({E}_{\mathrm{iso}})$	no evolution	BPL	17.2 (126)	−0.35 ± 0.01	−1.29 ± 0.12	1.80 ± 0.05
$\psi ({E}_{\mathrm{iso}})$	no evolution	CPL	15.4 (127)	−0.32 ± 0.01	⋯	2.63 ± 0.04

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The derived $\psi ({L}^{{\prime} })$ and $\psi ({E}^{{\prime} })$ correspond to the present-time GRB luminosity and energy distributions (at z = 0) and hence the local LF and EF in the comoving frame are roughly estimated as $\psi {({L}^{{\prime} })(1+z)}^{{\delta }_{L}}$ and $\psi {({E}^{{\prime} })(1+z)}^{{\delta }_{{\rm{E}}}}$, correspondingly. Taking into account that the z–${L}_{\mathrm{iso}}$ and z–${E}_{\mathrm{iso}}$ evolutions are established at $\lt 2\sigma$, the LF and EF calculated without accounting for the evolution, $\psi ({L}_{\mathrm{iso}})$ and $\psi ({E}_{\mathrm{iso}})$, may be of interest. We estimated these functions by setting ${\delta }_{L}$ and ${\delta }_{E}$ to zero, and found them very similar in shape to the present-time LF and EF (Figure 10). The results of the BPL and CPL fits to $\psi ({L}_{\mathrm{iso}})$ and $\psi ({E}_{\mathrm{iso}})$ are given in the last four lines of Table 7.

Finally, using the EP method, we estimated the cumulative GRB number distribution $\psi (z)$ and the derived GRBFR per unit time per unit comoving volume $\rho (z)$ (see the Appendix for the details). In Figure 11, we compare the star formation rate (SFR) data from the literature (Hopkins 2004; Hanish et al. 2006; Thompson et al. 2006; Li 2008; Bouwens et al. 2011) with GRBFRs derived from different z–L and z–E distributions. The GRBFR estimated from the evolution-corrected z–$L^{\prime}$ distribution exceeds the SFR at $z\lt 1$ and nearly traces the SFR at higher redshifts; the same behavior is noted for the GRBFRs estimated using both the evolution-corrected z–$E^{\prime}$ and the non-corrected z–${E}_{\mathrm{iso}}$ distributions. The low-z GRBFR excess over SFR is in agreement with the results reported in Yu et al. (2015) and Petrosian et al. (2015). Meanwhile, the only GRBFR that traces the SFR in the whole KW GRB redshift range is the $\rho (z)$ derived from the z–${L}_{\mathrm{iso}}$ distribution (i.e., not accounting for the luminosity evolution). Such behavior is known, e.g., from Wu et al. (2012), albeit for the GRBFR estimated from the z–${E}_{\mathrm{iso}}$ distribution.

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Figure 12 shows ${E}_{{\rm{p}},i}$ as a function of the burst durations T₉₀ in the observer and rest frames. In the observer frame the KW Type I GRBs are typically harder and shorter than Type II bursts, which is consistent with the classification obtained from the hardness–duration distribution (Figure 1), and this tendency shows no dependence on the burst redshift.

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In the cosmological rest frame, this pattern remains practically unchanged for GRBs at $z\lesssim 1.7$ but it appears to be less distinct when the whole sample is considered. Although in the rest frame Type I GRBs are still shorter than Type II GRBs, their rest-frame E_p, clustered around 1 MeV, are superseded by those of a significant fraction of the Type II population. The notable exceptions here are GRB 090510 and GRB 160410A, whose rest-frame peak energies exceed those of even the highest-z Type II GRBs. We note, however, that the derived rest-frame durations are affected by a variable energy-dependant factor (Section 4.1) and the KW rest-frame E_p are subject to the observational bias (Section 5.3) thus an interpretation of the rest-frame hardness–duration distribution should be done with care.

Using the data described in the previous sections, we tested KW GRB characteristics against ${E}_{{\rm{p}},i}$–S and ${E}_{{\rm{p}},{\rm{p}}}$–${F}_{\mathrm{peak}}$ correlations in the observer frame, and ${E}_{{\rm{p}},i,z}$–${E}_{\mathrm{iso}}$ ("Amati") and ${E}_{{\rm{p}},{\rm{p}},z}$–${L}_{\mathrm{iso}}$ ("Yonetoku") correlations in the rest frame, along with their collimated versions ${E}_{{\rm{p}},i,z}$–E_γ and ${E}_{{\rm{p}},{\rm{p}},z}$–L_γ.

To probe the existence of correlations, we calculated the Spearman rank-order correlation coefficients (${\rho }_{S}$) and the associated null-hypothesis (chance) probabilities or p values (${P}_{{\rho }_{S}};$ Press et al. 1992). The null hypothesis is that no correlation exists; therefore, a small p value indicates a significant correlation. It was shown that the Nukers' estimate is an unbiased slope estimator for the linear regression (Tremaine et al. 2002). The Nukers' estimate is based on minimizing:

$$\begin{eqnarray*}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{({y}_{i}-{{ax}}_{i}-b)}^{2}}{{a}^{2}{\sigma }_{{xi}}^{2}+{\sigma }_{{yi}}^{2}+{\sigma }_{\mathrm{int}}^{2}},\end{eqnarray*}$$

where ${\sigma }_{{xi}}^{2}$ and ${\sigma }_{{yi}}^{2}$ are the measurement errors; thus both variables are treated symmetrically in terms of their errors and there is no need to choose dependent and independent variables. Although a correlation may be highly significant, the reduced statistic, ${\chi }_{r}^{2}$, may be $\gg 1$ indicating that either the uncertainties are underestimated or there is an intrinsic dispersion in the correlation. To account for the intrinsic dispersion, an additional term, ${\sigma }_{\mathrm{int}}^{2}$, is added to the denominator and, in this case, ${\chi }_{r}^{2}$ is adjusted to ensure ${\chi }_{r}^{2}=1$. Therefore, we approximated a linear regression between $\mathrm{log}$-energy and $\mathrm{log}$ E_p using two methods, without ${\sigma }_{\mathrm{int}}$ and with the intrinsic scatter.

Table 8 summarizes the correlation parameters we obtained for subsamples of Type I GRBs, Type II GRBs, and Type II GRBs with ${t}_{\mathrm{jet}}$ estimates. The first column presents the correlation. The next three columns provide the number of bursts in the fit sample, ${\rho }_{S}$, and ${P}_{{\rho }_{S}}$. The next columns specify the slopes (a), the intercepts (b), and ${\sigma }_{\mathrm{int}}$. Since zeroing the intrinsic scatter yields ${\chi }_{r}^{2}\gg 1$ for all the subsamples (and that confirms the relevance of accounting for ${\sigma }_{\mathrm{int}}$), their values are of little interest and we do not present the fit statistics in the table.

Table 8.  Hardness–Intensity Correlations

Correlation	N	${\rho }_{S}$	${P}_{{\rho }_{S}}$	a	b	${a}_{{\sigma }_{\mathrm{int}}}$	${b}_{{\sigma }_{\mathrm{int}}}$	${\sigma }_{\mathrm{int}}$
Type I GRBs
${E}_{{\rm{p}},i}$ versus S	12	0.74	5.8 × 10−3	0.408 ± 0.043	4.98 ± 0.22	0.496 ± 0.117	5.52 ± 0.62	0.135
${E}_{{\rm{p}},{\rm{i}},z}$ versus ${E}_{\mathrm{iso}}$	12	0.83	9.5 × 10−4	0.364 ± 0.030	−15.70 ± 1.53	0.266 ± 0.068	−10.61 ± 3.47	0.181
${E}_{{\rm{p}},{\rm{p}}}$ versus ${F}_{\mathrm{peak}}$	12	0.54	7.1 × 10−2	0.340 ± 0.045	4.39 ± 0.19	0.349 ± 0.161	4.52 ± 0.74	0.188
${E}_{{\rm{p}},{\rm{p}},z}$ versus ${L}_{\mathrm{iso}}$	12	0.67	1.7 × 10−2	0.396 ± 0.034	−17.68 ± 1.78	0.243 ± 0.078	−9.61 ± 4.07	0.200
Type II GRBs
${E}_{{\rm{p}},i}$ versus S	137	0.59	3.7 × 10−14	0.418 ± 0.002	4.06 ± 0.01	0.295 ± 0.031	3.66 ± 0.14	0.227
${E}_{{\rm{p}},{\rm{i}},z}$ versus ${E}_{\mathrm{iso}}$	137	0.70	1.4 × 10−21	0.469 ± 0.003	−22.35 ± 0.14	0.338 ± 0.026	−15.27 ± 1.37	0.229
${E}_{{\rm{p}},{\rm{p}}}$ versus ${F}_{\mathrm{peak}}$	136	0.58	2.2 × 10−13	0.453 ± 0.004	4.68 ± 0.02	0.363 ± 0.041	4.31 ± 0.21	0.253
${E}_{{\rm{p}},{\rm{p}},z}$ versus ${L}_{\mathrm{iso}}$	136	0.73	1.6 × 10−23	0.494 ± 0.005	−23.32 ± 0.26	0.347 ± 0.029	−15.52 ± 1.51	0.251
Type II GRBs with ${t}_{\mathrm{jet}}$ estimates
${E}_{{\rm{p}},{\rm{i}},z}$ versus ${E}_{\mathrm{iso}}$	30	0.82	4.1 × 10−08	0.536 ± 0.004	−27.34 ± 0.21	0.418 ± 0.053	−19.62 ± 2.82	0.233
${E}_{{\rm{p}},{\rm{i}},z}$ versus ${E}_{\gamma }$	30	0.76	1.1 × 10−06	0.604 ± 0.008	−27.93 ± 0.42	0.499 ± 0.077	−22.69 ± 3.90	0.266
${E}_{{\rm{p}},{\rm{p}},z}$ versus ${L}_{\mathrm{iso}}$	30	0.75	1.5 × 10−06	0.529 ± 0.008	−25.12 ± 0.43	0.373 ± 0.063	−16.91 ± 3.30	0.282
${E}_{{\rm{p}},{\rm{p}},z}$ versus ${L}_{\gamma }$	30	0.61	3.1 × 10−04	0.731 ± 0.016	−33.87 ± 0.78	0.376 ± 0.097	−16.14 ± 4.86	0.343

Note. N is the number of bursts in the fit sample, ${\rho }_{S}$ is the Spearman correlation coefficient, ${P}_{{\rho }_{S}}$ is the corresponding chance probability, a (${a}_{{\sigma }_{\mathrm{int}}}$) and b (${b}_{{\sigma }_{\mathrm{int}}}$) are the slope and the intercept for the fits without (with) intrinsic scatter ${\sigma }_{\mathrm{int}}$.

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For the subsamples of Type I and Type II KW GRBs, both the Amati and Yonetoku correlations improve considerably when moving from the observer frame to the GRB rest frame (${\rm{\Delta }}{\rho }_{S}\geqslant 0.1$), with only marginal changes in the slopes. We found the rest-frame correlations for Type II bursts to be the most significant, with ${P}_{{\rho }_{S}}\lt 2\times {10}^{-21}$. The derived slopes of the Amati and Yonetoku relations for those GRBs are very close to each other, 0.469 (${\rho }_{S}=0.70$, 138 GRBs) and 0.494 (${\rho }_{S}=0.73$, 137 GRBs), respectively. These values are in agreement with the original results of Amati et al. (2002) and Yonetoku et al. 2004 and their further improvements (e.g., Nava et al. 2012). When accounting for the intrinsic scatter, these slopes change to a more gentle ∼0.35 (with ${\sigma }_{\mathrm{int}}\sim 0.24$).

As one can see in Figure 13, the lower boundaries of both the Amati and Yonetoku relations are defined by GRBs with moderate-to-high detection significance, so the instrumental biases do not affect the correlations from this edge of the distributions. Meanwhile, all outliers in the relations lie above the upper boundaries of the 90% prediction intervals (PIs) of the relations. Since these bursts were detected at lower significance, with the increased number of GRB redshift observations, one could expect a "smear" of the hardness–intensity correlations due to more hard-spectrum/less-energetic GRB detections. Thus, using the KW sample, we confirm a finding of Heussaff et al. (2013) that the lower right boundary of the Amati correlation (the lack of luminous soft GRBs) is an intrinsic GRB property, while the top left boundary may be due to selection effects. This conclusion may also be extended to the Yonetoku correlation.

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The collimated versions of these relations were tested on the subsample of 30 Type II GRBs with reliable ${t}_{\mathrm{jet}}$ (last four lines of Table 8). We found that accounting for the jet collimation for the KW sample neither improves the significance of the correlations nor reduces the dispersion of the points around the best-fit relations. The slopes we obtained for the collimated Amati and Yonetoku relations are steeper compared to those of the non-collimated versions.

The ${E}_{{\rm{p}},i,z}$–${E}_{\mathrm{iso}}$ and ${E}_{{\rm{p}},{\rm{p}},z}$–${L}_{\mathrm{iso}}$ correlations for 12 Type I bursts are less significant when compared to those for Type II GRBs, and they are characterized by less steep slopes (0.364 and 0.396 for ${E}_{{\rm{p}},i,z}$–${E}_{\mathrm{iso}}$ and ${E}_{{\rm{p}},{\rm{p}},z}$–${L}_{\mathrm{iso}}$, respectively). It should be noted, however, that the rest-frame ${E}_{{\rm{p}},i}$ of Type I GRBs shows only a weak (if any) dependence on the burst energy below ${E}_{\mathrm{iso}}\sim {10}^{52}$ erg (Figure 13), and the same is true for the ${E}_{{\rm{p}},{\rm{p}},z}$–${L}_{\mathrm{iso}}$ relation at ${L}_{\mathrm{iso}}\lesssim 5\times {10}^{52}$ erg s⁻¹. Above these limits the slopes of both relations for Type I GRBs are similar to those for Type II GRBs. As one can see from the figure, all KW Type I bursts are hard-spectrum/low-isotropic-energy outliers in the Amati relation for Type II GRBs. In the ${E}_{{\rm{p}},{\rm{p}},z}$–${L}_{\mathrm{iso}}$ plane, this pattern is less distinct; at luminosities above ${L}_{\mathrm{iso}}\sim {10}^{52}$ erg s⁻¹ the Type I bursts nearly follow the upper boundary of the Type II GRB Yonetoku relation. Finally, the two KW Type I GRBs with available collimation data lie above 90% PI for the Type II GRB ${E}_{{\rm{p}},i,z}-{E}_{\gamma }$ relation and, simultaneously, within the 68% PI for the ${E}_{{\rm{p}},{\rm{p}},z}$–L_γ relation (Figure 13, lower panels).

We also calculated the collimation-corrected energetics for the ultraluminous KW GRB 110918A using ${t}_{\mathrm{jet}}=0.2\pm 0.13$ days estimated by Frederiks et al. (2013) from an extrapolation of early γ-ray/late X-ray afterglow data. As can be seen in Figure 13, the implied ${E}_{\gamma }\approx 1.1\times {10}^{51}$ erg and ${L}_{\gamma }\approx 1.9\times {10}^{51}$ erg s⁻¹ nicely agree with both hardness–intensity relations for our "reliable ${t}_{\mathrm{jet}}$" GRB sample. This supports the correctness of the ${t}_{\mathrm{jet}}$ estimate and favors the conclusion of Frederiks et al. (2013) that a tight collimation of the jet (${\theta }_{\mathrm{jet}}\sim 1\buildrel{\circ}\over{.} 6$) must have been a key ingredient to produce this unusually bright burst.